Heat flow of Yang-Mills-Higgs functionals in dimension two

TL;DR

This paper studies the heat flow of Yang-Mills-Higgs functionals on Riemannian surfaces, establishing global weak solutions, analyzing singularities, and describing long-term behavior.

Contribution

It proves existence of global weak solutions for the heat flow of Yang-Mills-Higgs functionals on surfaces and characterizes their singularities and asymptotic behavior.

Findings

Existence of global weak solutions for any initial data in H^1.

Finite number of singularities where solutions are not smooth.

Energy identity and asymptotic behavior at infinity.

Abstract

We consider the heat flow of Yang-Mills-Higgs functional where the base manifold is a Riemannian surface and the fiber is a compact symplectic manifold. We show that the corresponding Cauchy problem admits a global weak solution for any $H^1$-initial data. Moreover, the solution is smooth except finitely many singularities. We prove an energy identity at finite time singularities and give a description of the asymptotic behavior at time infinity.